Chain Ladder Method

Study guide: Exam 5

Chain Ladder Method (development or link-ratio method) projects each cohort to ultimate by multiplying its latest cumulative amount by a cumulative development factor derived from the historical pattern.

$U_{\text{AY}} = C_{\text{AY},\,n} \times \text{CDF}_{n \to \text{ult}}$

$\text{IBNR} = U_{\text{AY}} - C_{\text{AY},\,n}$

The procedure: build the cumulative triangle → compute age-to-age factors → select a factor for each interval → chain them into CDFs with a tail → apply to the latest diagonal.

The assumptions, and each is a way the method fails:

Each cohort develops in the same pattern. Violated by a change in mix, limits or coverage.
The pattern is stable over calendar time. Violated by a change in case adequacy, settlement speed or inflation — the diagonal effects.
Future development is proportional to what has emerged. This is the strong one: the method takes whatever sits on the diagonal at face value and scales it. A shock loss reported early is multiplied by the full CDF.

Further points:

The chain ladder is at its best on mature cohorts with stable patterns and ample volume — and at its worst at immature ages in long-tail lines, where the CDF is large and the diagonal is thin. There, BF or the expected claims technique is preferred.
Run it on paid and reported data both. The two answers rest on the same claims, so their difference is a diagnostic about case adequacy and settlement rates, not a menu to average.
It is entirely data-driven: nothing external — premium, exposure, rate level or judgment about the year — enters the estimate. That independence is its strength as a check on other methods and its weakness as a standalone estimate.

Chain Ladder Projection Across Three Years Cumulative reported losses ($000s):

AY 12 mo 24 mo 36 mo

$2022$ $575$ $875$ $1{,}025$

$2023$ $600$ $900$

$2024$ $625$

Selected factors: $f_{12\text{–}24} = 1.500$ , $f_{24\text{–}36} = 1.170$ , tail $= 1.081$ .

Project each year and compute total IBNR.
Answer
 $\begin{align*} \text{CDF}_{36} &= 1.081 \\ \text{CDF}_{24} &= 1.170 \times 1.081 = 1.265 \\ \text{CDF}_{12} &= 1.500 \times 1.265 = 1.898 \end{align*}$ 
AY Reported CDF Ultimate IBNR

$2022$ $1{,}025$ $1.081$ $1{,}108$ $83$

$2023$ $900$ $1.265$ $1{,}139$ $239$

$2024$ $625$ $1.898$ $1{,}186$ $561$

Total $2{,}550$ $3{,}433$ $883$

Total IBNR is $\$883{,}000$ , of which $64\%$ sits on the single most recent year. That concentration is the signature of the method: the youngest cohort carries the largest factor, the thinnest data and almost all the uncertainty.

AY	12 mo	24 mo	36 mo
$2022$	$575$	$875$	$1{,}025$
$2023$	$600$	$900$
$2024$	$625$

AY	Reported	CDF	Ultimate	IBNR
$2022$	$1{,}025$	$1.081$	$1{,}108$	$83$
$2023$	$900$	$1.265$	$1{,}139$	$239$
$2024$	$625$	$1.898$	$1{,}186$	$561$
Total	$2{,}550$		$3{,}433$	$883$

Where the Chain Ladder Breaks Accident year $2024$ at $12$ months has reported losses of $\$4{,}500{,}000$ , including one claim reserved at its $\$1{,}500{,}000$ policy limit. The selected $\text{CDF}_{12}$ is $3.20$ , built from years containing no such claim. Earned premium is $\$14{,}000{,}000$ and the a priori loss ratio is $65\%$ .

Estimate ultimate losses.
Answer Chain ladder as-is:

$\$4{,}500{,}000 \times 3.20 = \$14{,}400{,}000$

— a $103\%$ loss ratio, against an a priori of $65\%$ , driven by one claim. The method has implicitly forecast that the $\$1.5$ M claim will grow to $\$4.8$ M, which it cannot: it is already at policy limits.

Corrected — remove the shock loss, develop the rest, add it back:
 $\begin{align*} \text{Ordinary} &= \$3{,}000{,}000 \times 3.20 = \$9{,}600{,}000 \\ \text{Known large claim} &= \$1{,}500{,}000 \\ \text{Provision for further large claims} &\approx \$700{,}000 \\[4pt] \text{Ultimate} &= \$11{,}800{,}000 \end{align*}$ 
— an $84\%$ loss ratio, still adverse but plausible.

BF as a cross-check, using the corrected treatment: with $p = 1/3.20 = 31.25\%$ ,

$\$4{,}500{,}000 + 0.6875 \times 0.65 \times \$14{,}000{,}000 = \$10{,}756{,}000$

The three answers span $\$3.6$ M. The chain ladder's $\$14.4$ M is the one that should be discarded outright — not because the method is wrong, but because its proportionality assumption is being applied to a diagonal that violates it.